Berry-Phase-Induced Dimerization in One-Dimensional Quadrupolar Systems

We investigate the effect of the Berry phase on quadrupoles that occur, for example, in the low-energy description of spin models. Specifically, we study here the one-dimensional bilinear-biquadratic spin-one model. An open question for many years about this model is whether it has a nondimerized fluctuating nematic phase. The dimerization has recently been proposed to be related to Berry phases of the quantum fluctuations. We use an effective low-energy description to calculate the scaling of the dimerization according to this theory and then verify the predictions using large scale density-matrix renormalization group simulations, giving good evidence that the state is dimerized all the way up to its transition into the ferromagnetic phase. We furthermore discuss the multiplet structure found in the entanglement spectrum of the ground state wave functions.

Figure 1

(a) Phase diagram of the Hamiltonian (1). (b) In the quadrupolar state $|y⟩$ the spin fluctuates over the light yellow region perpendicular to the director, shown as dark green ellipsoid [14, 15]. (c) Rotating a director in the projective plane, a Berry phase is picked up on the red path only (the dotted red line connects equivalent points in ${\mathbb{R}\mathbb{P}}^2$). (d) World lines of directors in $1+1$-dimensional space time. The directors tend to align with adjacent ones in space (due to the spring constant $K$), and they can rotate in time, with a moment of inertia $I$. Rotations of a director as a function of time lead to a Berry phase. Directors can either rotate by an angle 0 or $\pi$, as shown in the two highlighted world lines, picking up a phase $\pi$ in the latter case. To keep track of the total rotation, we add a red dot to the green ellipsoid. In between domains of different windings, $\pi$ vortices appear (black dots).

Figure 2

(a) The correlation length $\ln \xi$ and (b) the bulk dimerization $\ln \mathcal{D}$ as a function of $1/\sqrt{\mathrm{\Delta }\theta }$ and $1/\mathrm{\Delta }\theta$, respectively. The data are extrapolated to the infinite-$m$ limit (blue open circles with error bars). Red fitting lines: $\ln \xi =-2.180+3.596/\sqrt{\mathrm{\Delta }\theta }$ in (a) and $\ln \mathcal{D}=2.054-1.047/\mathrm{\Delta }\theta$ in (b).

Figure 3

Three different regions can be distinguished by the scaling of the (a) entanglement entropy $S_E$ and (b) bipartite spin fluctuation $\mathcal{F}$ as a function of $\ln L$ for different values of $\mathrm{\Delta }\theta /\pi$. Converged (interpolated) data are marked by filled (open) symbols. Fat green, yellow, and red lines indicate the expected asymptotic behavior in the three regimes (see text for details).

Figure 4

Multiplet structures of the ES as a function of $\mathrm{\Delta }\theta$ for (a) even and (b) odd $L/2$. Levels originating from the SU(3) point are colored and their multiplicity indicated. (c) and (d) ES as a function of $S_{\mathcal{L}}(S_{\mathcal{L}}+1)$ for $\mathrm{\Delta }\theta /\pi =0.05$. Black dashed lines, $A_0+A_1{S}_{\mathcal{L}}(S_{\mathcal{L}}+1)$, are fitting to the lower edge of the spectrum. Panels (a)-(d) are obtained with $m=4000$. (e) The slope of tower edges as a function of $1/\mathrm{\Delta }\theta$. Each slope value is extrapolated to the infinite-$m$ limit, with error bars less than the symbol size. The red solid line gives $1/A_1=0.745+0.432/\mathrm{\Delta }\theta$. (f) Double logarithmic plot, with the green dashed line showing the expected asymptotic slope of $1/A_1\propto 1/\mathrm{\Delta }\theta$.